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84
Signals and Spectra Chap. 2
is the phase response of the network. Furthermore, since h(t) is a real function of time (for real
networks), it follows from Eqs. (2–38) and (2–39) that ƒH(f)ƒ is an even function of frequency
and lH(f) is an odd function of frequency.
The transfer function of a linear time-invariant network can be measured by using a
sinusoidal testing signal that is swept over the frequency band of interest. For example, if
then the output of the network will be
x(t) = A cos v 0 t
y(t) = AƒH(f 0 )ƒ cos [v 0 t + lH(f 0 )]
(2–138)
where the amplitude and phase may be evaluated on an oscilloscope or by the use of a vector
voltmeter.
If the input to the network is a periodic signal with a spectrum given by
X(f) =
n=q
a
n=-q
c n d(f - nf 0 )
(2–139)
where, from Eq. (2–109), {c n } are the complex Fourier coefficients of the input signal, the
spectrum of the periodic output signal, from Eq. (2–134), is
Y(f) =
n=q
a
n=-q
c n H(nf 0 ) d(f - nf 0 )
(2–140)
We can also obtain the relationship between the power spectral density (PSD) at the input,
x ( f ), and that at the output, y (f), of a linear time-invariant network. † From Eq. (2–66),
we know that
y (f) = lim
T: q
Using Eq. (2–134) in a formal sense, we obtain
1
T ƒY T (f)ƒ2
(2–141)
y (f) = ƒH(f)ƒ 2
lim
T: q
1
T ƒX T (f)ƒ2
or
y (f) = |H(f)| 2 x (f)
(2–142)
Consequently, the power transfer function of the network is
G h (f) = y(f)
= |H(f)| 2
x (f)
A rigorous proof of this theorem is given in Chapter 6.
(2–143)
† The relationship between the input and output autocorrelation functions R x (t) and R y (t) can also be obtained
as shown by (6–82).